In the branches of abstract algebra known as ring theory and module theory, each right (resp. left) R-module M has a singular submodule consisting of elements whose annihilators are essential right (resp. left) ideals in R. In set notation it is usually denoted as . For general rings, is a good generalization of the torsion submodule tors(M) which is most often defined for domains. In the case that R is a commutative domain, .
If R is any ring, is defined considering R as a right module, and in this case is a two-sided ideal of R called the right singular ideal of R. The left handed analogue is defined similarly. It is possible for .
and 18 Related for: Singular submodule information
theory and module theory, each right (resp. left) R-module M has a singularsubmodule consisting of elements whose annihilators are essential right (resp...
associative algebra that is a simple ring. singularsubmodule The right (resp. left) R-module M has a singularsubmodule if it consists of elements whose annihilators...
Verma module by its maximal submodule. A Verma module is irreducible if and only if it has no singular vectors. A singular vector or null vector of a highest...
M additively. Given such a G-module M, it is natural to consider the submodule of G-invariant elements: M G = { x ∈ M | ∀ g ∈ G : g x = x } . {\displaystyle...
also belong to I {\displaystyle I} . (Equivalently, if it is a graded submodule of R {\displaystyle R} ; see § Graded module.) The intersection of a homogeneous...
residue field. essential 1. A submodule M of N is called an essential submodule if it intersects every nonzero submodule of N. 2. An essential extension...
order of the factors. (M2) The torsion submodule is a direct summand. That is, there exists a complementary submodule P {\displaystyle P} of M {\displaystyle...
such that every submodule of a module of finite type is also of finite type. Ideals of a ring R {\displaystyle R} are the submodules of R {\displaystyle...
y\rangle } is preserved. The root lattice of a root system Φ is the Z-submodule of E generated by Φ. It is a lattice in E. The group of isometries of E...
(L,B)} , such that L {\displaystyle L} is a free A {\displaystyle A} -submodule of M {\displaystyle M} such that M = Q ( q ) ⊗ A L ; {\displaystyle M=\mathbb...
principal ideal domain. In particular, over a principal ideal domain, every submodule of a free module is free, and the fundamental theorem of finitely generated...
for the divisor class of d. If F , F ′ {\displaystyle F,F'} are maximal submodules of M, then c ( χ ( M / F ) ) = c ( χ ( M / F ′ ) ) {\displaystyle c(\chi...
W_{\lambda }.} W λ {\displaystyle W_{\lambda }} contains a unique maximal submodule, and its quotient is the unique (up to isomorphism) irreducible representation...
{\displaystyle M_{s_{\alpha }\cdot \lambda }} is isomorphic to a unique submodule of M λ {\displaystyle M_{\lambda }} . In (1), we identified M s α ⋅ λ...
the statement: There exists a finitely generated A {\displaystyle A} -submodule of B {\displaystyle B} that contains A [ b ] {\displaystyle A[b]} . Finally...
generalized to free modules over a principal ideal domain. For instance, submodules of free modules over principal ideal domains are free, a fact that Hatcher...
{C}}\Omega ^{i}({\mathcal {O}})\subseteq \Omega ({\mathcal {O}})} be the submodule of differential forms over O {\displaystyle {\mathcal {O}}} whose restriction...
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